Journal of Statistical Mechanics: Theory and Experiment
Volume 2006
February 2006
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P I Hurtado et al J. Stat. Mech. (2006) P02004 doi:10.1088/1742-5468/2006/02/P02004
Understanding scale invariance in a minimal model of complex relaxation phenomena
P I Hurtado1, J Marro and P L Garrido
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| Figure 1. Typical evolution in which the magnetization is observed to decay by jumps to the final stable state. This is for a single particle of radius R = 30 ( ~ 103 spins) at low temperature, and for small values of h and p (see the main text). The time axis shows t - τ0 in MCS (Monte Carlo steps per site) with τ0 = 1030 MCS; this is of the order of the duration of the initial metastable state. The inset shows a significant detail of the relaxation. |
| Figure 2. Log-log plot of the size distribution P(Δm) of large avalanches for an ensemble of independent particles of radius (from bottom to top) R = 30, 42, 60, 84 and 120, respectively. Plots of the duration distribution P(Δt) versus cΔtγ for each R are also shown (×), with c ≈ 0.5 and γ ≈ 1.52 (see the text). For visual convenience, the curves are shifted vertically by 4n with n = 0 to 4 from bottom to top. Running averages have been performed for clarity purposes. |
Figure 3. Log-log plot of the duration distribution
P(Δt) for the same ensembles of particles as in figure 2. For visual convenience, the curves are shifted vertically by
2n with n = 0 to 4 from bottom to top. Running averages have been performed for clarity purposes.
Inset: log-log plot of the size (top) and duration (bottom) cut-offs  versus R. Lines are power-law fits. |
Figure 4. Semilogarithmic plot of  (solid line) and  (dotted line) as a function of magnetization, after averaging over
3500 independent runs. Notice the non-trivial structure uncovering a high degree of
correlation between the mean size of avalanches, , and the average curvature of the interface at which the avalanche originates. |
Figure 5. Semilogarithmic plot of P(Δm|C), the size distribution for avalanches developing at a wall of constant curvature,
C;
C increases from bottom to top. Here,  . (For visual convenience, the curves are shifted vertically by
10n with n = 0 to 4 from bottom to top.) Running averages have been performed for clarity purposes. |
| Figure 6. Solid lines are predictions from equation (8) for n = 200, Amin = 0.007 and Amax = 1. The symbols stand for the avalanche duration (lower curve) and size (upper curve) when R = 60, i.e., two of the data sets in figures 2 and 3. In this particular case, the finite-size exponent is τ(R = 60) = 2.06(2), allowing direct comparison with equation (8). |
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